3.2.3—Log equations

Solve logarithmic equations by combining legal logs, converting to an exponential equation and checking every solution in the original domain.

If a quadratic appears after exponentiating, both algebraic roots are only candidates; reject any that make a log argument zero or negative.

log₃(x)+log₃(x−2)=1 requires x>2. It becomes x(x−2)=3, whose roots are 3 and −1; only x=3 survives.

Exponentiating an equation preserves equivalence only when both sides were defined; it cannot legalise an invalid log argument.